It has been observed that, given an algebraic quantum field theory (AQFT) on a manifold M and an open cover <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2024_1529_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\{M_\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of M, it is typically not possible to recover the global algebra of observables on M by simply gluing the underlying local algebras subordinate to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2024_1529_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\{M_\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. Instead of gluing local algebras, we introduce a gluing construction for AQFTs subordinate to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2024_1529_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\{M_\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and we show that for simple examples of AQFTs, constructed out of geometric data, gluing the local AQFTs subordinate to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2024_1529_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\{M_\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>M</mi> <mi>α</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> recovers the global AQFT on M.

Gluing Algebraic Quantum Field Theories on Manifolds

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2025年1月8日

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Angelos Anastopoulos,
Marco Benini

摘要

It has been observed that, given an algebraic quantum field theory (AQFT) on a manifold M and an open cover $\{M_\alpha \}$ ${M_{α}}$ of M, it is typically not possible to recover the global algebra of observables on M by simply gluing the underlying local algebras subordinate to $\{M_\alpha \}$ ${M_{α}}$ . Instead of gluing local algebras, we introduce a gluing construction for AQFTs subordinate to $\{M_\alpha \}$ ${M_{α}}$ and we show that for simple examples of AQFTs, constructed out of geometric data, gluing the local AQFTs subordinate to $\{M_\alpha \}$ ${M_{α}}$ recovers the global AQFT on M.